How can you understand the particle in a box application of schrodingers equation in your physical world?

Answer 1

Well the particle in a box problem, which is described as a particle that is in a zero potential well whose walls have infinite potential. This would be like falling down a mine shaft with no way of getting out. Now what this problem helps us to understand is the probability attributes of quantum mechanics. We can find the expectation value for the position, in Dirac notation <psi | x | psi>, which will say that the most probable place that the particle is in the middle of the well (or box). Now just as an example, grab a marble and a cereal bowl (it must have a curved bottom). Now roll the marble down the side of bowl. Now glance at the bowl and look away. Make a note of where the marble is then do this a bunch of times and keep track of where the marble was every time you looked at it. If you were to plot these results you will find that even though the marble is always moving it spends most of it's time near the center of the bowl. Thus, you can accurately state that though the marble is not always at the center of the bowl the probability of it being near the center when you measure it's position (look at it) is higher than the probability of it being near the edges of the bowl! That is essentially what the particle in a box (infinite square well) problem is saying, with regards the the expectation value of position.

Sorry so long!